gravity anomalies of some simple shapes -...
TRANSCRIPT
Gravity anomalies of some simple shapes
g at surface subsurface density
subsurface geology
(observation) (material property)
GRAVITY EXPLORATION
Reminder: we are working with values about... 0.01-0.001 mgal ≈ 10-8 - 10-9 g
N !!!
Applications of Gravity Surveys • Exploration of fossil fuels (oil, gas, coal) • Exploration of bulk mineral deposit (mineral, sand, gravel) • Exploration of underground water supplies • Engineering/construction site investigation • Cavity detection • Glaciology • Regional and global tectonics • Geology, volcanology • Shape of the Earth, isostasy • Military (e.g., detection of underground bunkers)
Gravity and gravity anomalies
Density contrast (variation in subsurface
mass distribution relative to some “average” Earth)
Gravity anomaly (difference between observed gravity and expected gravity)
We computed the gravity anomaly of a spherical ore body at X (immediately on top of the ore body).
à How does the gravity anomaly vary laterally?
Gravity forward problems:
Total gravity
g + Δg
Δg
Δgx
Δgz
g
Δg
g
θ
Vertical gravity
g + Δgz
Also, Δg is usually small
• θ is small
• Δgx is small
à consider only anomalies in vertical gravity (for exploration studies…for tectonic studies, the horizontal component can be significant, e.g., in mountainous areas)
g+Δg
Instruments can only measure vertical component of gravity.
Vertical gravity anomaly for a point mass
2hmGg Δ
=Δ
For a point mass at distance r:
Δm is mass difference w.r.t. surrounding material
Vertical component is:
32zhmzGosθc
hmGg Δ
=Δ
=Δ
hz
ggosθc z=
Δ
Δ=Note:
Gravity anomaly of any body • divide body into “point masses”
• at each position
• calculate gravity anomaly due to each point mass
• add up all anomalies
Δg1 Δg2 Δg3 Δg4
Mathematically (in 3D): ( ) 'dz'dy'dxhz'zGg 3z−
ρΔ∫∫∫=Δ
3z
hmzGg Δ
=Δ
( ) ( ) ( )222 z'zy'yx'xh −+−+−=Where: & (x, y, z) is the observation point
Gravity anomaly of a buried sphere
Sphere is more dense than surrounding material – therefore has a mass excess (MS)
Can show that the vertical component of gravity will vary as:
€
gz(x) =GMSz
x 2 + z2( )32
NOTE: x is the horizontal distance from the center of sphere to the point of measurement (just like for a point mass)
SOME VALUES
z = 100 m
r = 50 m
Δρ =2000 kg m-3
MS ~ 109 kg
€
gz(x) =GMSz
x 2 + z2( )32
( )ρΔ×⎟⎠
⎞⎜⎝
⎛ π= 3S r
34M
Where:
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
€
gz(x) = __________
Gravity anomaly of a buried sphere
• maximum gz at point above the centre of the sphere (x=0 m)
• gz decreases rapidly away from the sphere
• maximum value of gz is:
2Smax
z zGM
g = -250 -200 -150 -100 -50 0 50 100 150 200 2500
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
Gravity anomaly of a buried sphere
• maximum gz at point above the centre of the sphere (x=0 m)
• gz decreases rapidly away from the sphere
• maximum value of gz is:
• half-width of the curve is where gz = gz
max / 2
• can show that x1/2=0.766 z
• equivalently: z = 1.305 x1/2
2Smax
z zGM
g = -250 -200 -150 -100 -50 0 50 100 150 200 2500
0.1
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
gzmax
X1/2
Gravity anomaly of a buried sphere
Forward problem:
Given z, r, and Δρ
◊ can calculate gz (x)
Inverse problem:
Observe gz (x) à find gzmax , x1/2
• can now determine depth:
z = 1.305 x1/2
• and MS: GzgM2max
zS =
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
gzmax
X1/2
Changing the depth of the sphere…
2Smax
z zGM
g =
x1/2 = 0.766 z
Deeper sphere
à broader, smaller amplitude gravity anomaly
(Lowrie, 2007)
Half-width technique (half maximum)
(Lowrie, 2007)
The width (wavelength) of the anomaly is related to the depth of the body that causes it.
Deeper body = wider anomaly
à measure the half-width (x1/2) of an anomaly to get the approximate depth of the body.
2D: cylinder z = x1/2
3D: sphere z = 1.305 x1/2
where z is the center.
- for other shapes, z represents the maximum depth of the top of the body
Buried horizontal cylinder
Cylinder with radius a is at depth z. The density contrast between the cylinder and surrounding material is Δρ.
Can show that the vertical component of gravity will vary as:
€
gz(x) =2Gπr2zΔρx 2 + z2( )
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
€
gz(x) =2Gπr2zΔρx 2 + z2( )
SOME VALUES
axis at x = 0 m
z = 100 m
r = 50 m
Δρ =2000 kg m-3
(same values that were used for the sphere)
• maximum gz is at the axis of the cylinder (x=0 m)
• maximum value of gz is:
• can show that half-width is related to depth of the centre of the cylinder:
x1/2= z
Important observations
zrG2g2
maxz
ρΔπ=
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
Changing the radius
Half width: x1/2 = z zrG2g2
maxz
ρΔπ=
Changing the depth
(see textbook for the parameter values used to generate these plots)
SPHERE: CYLINDER:
• why is gzmax larger for the cylinder?
• if you recorded these profiles in the field, how would you decide if the anomaly was due to a sphere or a cylinder?
Compare the gravity anomaly profiles:
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2
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
-250 -200 -150 -100 -50 0 50 100 150 200 2500
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
% a comparison between cynlinder and sphere x = linspace(-250,250,501); pi=3.1415926; G=6.67e-11; z=100; r=50; Vs=4*pi*r*r*r/3.0; drho = 2000; gz_sph=G*Vs*z*drho./(z*z+x.*x).^1.5*100000; gz_cyl=2*pi*G*r*r*drho*z./(z*z+x.*x)*100000; gz_sph_scaled=gz_sph.*2.0953/0.7; plot(x, gz_sph) hold on plot(x, gz_cyl, 'r') hold on plot(x, gz_sph_scaled, 'g') xlabel('x (m)') ylabel('gravity anomaly gz (mgal)') legend('sphere', 'cylinder', 'sphere scaled');
Simple code to try on matlab
Gravity profiles vs. maps SPHERE
PROFILE
MAP
CYLINDER
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
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-50
0
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150
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X distance (m)
Y di
stan
ce (m
)
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0
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X distance (m)
Y di
stan
ce (m
)
0
0.5
1
1.5
2
Example Problem Buried spherical iron ore body: • Centre of ore body:
x = ______ m
• Maximum value of gz:
gzmax = _____ mgal
• Half-width:
x1/2 = ______ m
• Depth of sphere:
z = _______m
• Excess mass of sphere:
MS = ______ kg
(G = 6.67 x 10-11 m3 kg-1 s-2)
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Distance (m)
Gra
vity
ano
mal
y, g
z (m
gal)
Is it possible to determine the radius or density contrast of the sphere?
Infinite horizontal layer
Can show that the gravitational attraction of this slab at P is:
zG2gz Δρπ=
à gz does not depend on the x-position of the measurement or on the depth of the layer -- WHY?
Infinite horizontal layer
à gz does not depend on the x-position of the measurement or the depth of the layer WHY?
zG2gz Δρπ=
Infinite Horizontal Layer
Can also think about this in terms of gravity anomaly caused by a density contrast:
zG2gz ΔρΔπ=Δ where Δρ=ρ0-ρ
Δz
Gravity value is: zG2gz Δρπ=
Both slabs have the same gz – can not tell them apart using only gravity data
à non-uniqueness
Compare the gravity response of these two layers:
Computer modeling of gravity anomalies
• for more complex geometries, gravity anomalies can be calculated using computer modeling programs
• models can be 2D or 3D
• for 2D: polygon is assumed to extend infinitely in the strike direction (in and out of page)
Δg1 Δg2 Δg3 Δg4
Verification of modeling program • before using any program, must verify that the calculations are accurate
• HOW? construct a model where the answer is known to see if the program gives you the right answer
• for example – buried horizontal cylinder
8-sided “cylinder”
line = model result dots = expected answer
Why is model gz less than expected?
BASIN 1 BASIN 2
• minimum gz is the nearly the same for both basins
• edge effects extend further into Basin 2
Δρ = -400 kg/m3 Δρ = -200 kg/m3